Conservation Laws & Applications
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1 Rocky Mountain Mathematics Consortium Summer School Conservation Laws & Applications Lecture V: Discontinuous Galerkin Methods James A. Rossmanith Department of Mathematics University of Wisconsin Madison July 2 st, 2010 J.A. Rossmanith RMMC /35
2 Outline 1 1D DG-FEM 2 2D Cartesian DG-FEM 3 2D mesh generation 4 2D unstructured DG-FEM J.A. Rossmanith RMMC /35
3 Outline 1 1D DG-FEM 2 2D Cartesian DG-FEM 3 2D mesh generation 4 2D unstructured DG-FEM J.A. Rossmanith RMMC /35
4 Basic idea Spatial discretization [Cockburn & Shu, 1990 s] Map each element to [ 1, 1]: Basis functions: φ (l) (ξ) = Orthogonality: Galerkin expansion: q h (t, x) x = x i + ξ x 2 j 1, ff 5 3 ξ, `3ξ 2 7 1, `5ξ 3 3ξ, Z 1 1 φ (l) (ξ) φ (k) (ξ) dξ = δ lk 2 1 T = nx Q (l) (t) φ (l) (ξ(x)) l=1 J.A. Rossmanith RMMC /35
5 Basic idea Cofficients are computed via L 2-projection: E Q (1) (t),..., Q (n) (t) = 1 Z " 2 1 nx q(t, ξ) Q (l) (t) φ (ξ)# (l) dξ 2 1 Z " E 1 Q = q(t, ξ) (k) 1 l=1 # nx Q (l) (t) φ (l) (ξ) φ (k) (ξ) dξ = 0 l=1 Q (k) (t) = 1 2 Z 1 1 q(t, ξ) φ (k) (ξ) dξ J.A. Rossmanith RMMC /35
6 Basic idea Weak formulation: j d 1 dt Z 1 1 Z 1 1 Q (k) i = 1 Z 1 f q h (t, ξ) x 1 Conservation: φ (l)n q,t + f(q),x o dξ = 0 ff φ (l) q dξ = 1 Z 1 φ (l) f(q),ξ dξ x 1 nx q(t, ξ) Q (k) (t) φ (k) (ξ) Ti «k = 1 = Time integrate = h k=1 φ (k) 1,ξ (ξ) dξ»φ (k) (1) F x i+ φ (k) ( 1) F 12 i 12 (1) Q i = 1»F x i+ F 12 i 12 i n+1 h i» n = Q (1) t i F x i+ F 12 i 12 Q (1) i J.A. Rossmanith RMMC /35
7 Basic idea Weak formulation: j d 1 dt Z 1 1 Z 1 1 Q (k) i = 1 Z 1 f q h (t, ξ) x 1 Conservation: φ (l)n q,t + f(q),x o dξ = 0 ff φ (l) q dξ = 1 Z 1 φ (l) f(q),ξ dξ x 1 nx q(t, ξ) Q (k) (t) φ (k) (ξ) Ti «k = 1 = Time integrate = h k=1 φ (k) 1,ξ (ξ) dξ»φ (k) (1) F x i+ φ (k) ( 1) F 12 i 12 (1) Q i = 1»F x i+ F 12 i 12 i n+1 h i» n = Q (1) t i F x i+ F 12 i 12 Q (1) i J.A. Rossmanith RMMC /35
8 Basic idea Time-stepping: Space-time methods (implicit) Lax-Wendroff (Taylor-series + replace time by spatial derivatives) Total variation diminishing Runge-Kutta: U = U n + t L(U n ) U = 3 4 U n U t L(U ) U n+1 = 1 3 U n U t L(U ) Numerical fluxes: Can use either exact or approximate Riemann solvers Unlike WP-FVM, order is not tied to Riemann solver Only need F ( q, q) = f( q) Local Lax-Friedrichs: F (Q r, Q l ) = 1 2 (f(q l) + f(q r)) 1 2 s (Qr Q l), s = max λ(f,q) J.A. Rossmanith RMMC /35
9 Basic idea Time-stepping: Space-time methods (implicit) Lax-Wendroff (Taylor-series + replace time by spatial derivatives) Total variation diminishing Runge-Kutta: U = U n + t L(U n ) U = 3 4 U n U t L(U ) U n+1 = 1 3 U n U t L(U ) Numerical fluxes: Can use either exact or approximate Riemann solvers Unlike WP-FVM, order is not tied to Riemann solver Only need F ( q, q) = f( q) Local Lax-Friedrichs: F (Q r, Q l ) = 1 2 (f(q l) + f(q r)) 1 2 s (Qr Q l), s = max λ(f,q) J.A. Rossmanith RMMC /35
10 Gaussian quadrature Need to evaluate integrals of the form I = Z 1 1 f(ξ) dξ Gaussian quadrature has maximum degree of precision: 2n 1 Gaussian quadrature rules: I 1 = 2f(0) I 2 = X f ± 1 «3 ± I 3 = 8 9 f (0) + 5 X «3 f ± 9 5 When integrating against φ (l) need rule with N points When integrating against φ (l),ξ ± need rule with N 1 points J.A. Rossmanith RMMC /35
11 Moment limiters [Krivodonova, 2007] Moment representation: Q (1), Q (2), Q (3),... Moment limiter: = minmod Q (k) i Q (2) i. = minmod Q (k) i, c (k) Q (k 1) i Q (k 1) i 1, c (k) Q (k 1) i+1 Q (k 1) i Q (2) i, c (2) Q (1) i Q (1) i 1, c (2) Q (1) i+1 Q(1) i J.A. Rossmanith RMMC /35
12 Artificial viscosity [Persson & Peraire, 2006] q,t + f(q),x = (ε B(q) v),x v = q,x Cell-by-cell smoothness indicator: q h = NX Q (l) φ (l), ˆq h = l=1 N 1 X l=1 Q (l) φ (l) S := qh ˆq h, q h ˆq h, S x 2N (smooth solutions) q h, q h ( 0 if S < TOL ε := ε 0 if S TOL Local DG method for elliptic & parabolic PDEs: Multiply both equations by same test function, φ (l), and 1 int-by-parts Difficulties: time-step restriction due to parabolic part, parameters J.A. Rossmanith RMMC /35
13 Discontinuous Galerkin FEM Why? Comparison of properties: FVM FDM FEM DG High-order Unstructured meshes Stability for conservation laws Explicit time-stepping ( ) Elliptic equations ( ) Difficulties with DG-FEM No universally accepted limiters Generally small time-restrictions due to sub-grid resolution Errors in replacing exact integrals by Gaussian quadrature Entropy stability and explicit time-stepping J.A. Rossmanith RMMC /35
14 Discontinuous Galerkin FEM A brief history First introduced by [Reed & Hill, 1973] for neutron transport: σq + (uq) = f First analysis by [Lesaint & Raviart, 1974] showing O(h N ) Sharp analysis [Johnson, 1986] showing O(h N+ 1 2 ) 1990 s: Extension to general conservation laws [Cockburn & Shu] Out of all this came RKDG (Runge-Kutta discontinuous Galerkin) Last 15 years: high-order, adaptivity, nodal/modal,...: Hesthaven, Warburton, Karniadakis, Kopriva, Shu, Persson, Munz,... Applications: Fluid dynamics Aero-acoustics Maxwell equations Plasma physics Shallow water waves J.A. Rossmanith RMMC /35
15 Outline 1 1D DG-FEM 2 2D Cartesian DG-FEM 3 2D mesh generation 4 2D unstructured DG-FEM J.A. Rossmanith RMMC /35
16 DG-FEM on Cartesian grids Map each element to canonical element [ 1, 1] [ 1, 1] x = x i + ξ x 2 y = y j + η y 2 Basis functions (up to third-order accuracy): φ (l) = n 1, Orthonormality: D φ (m), φ (n)e := 1 4 Galerkin approximation: 3 ξ, 3 η, 3 ξη, 5 2 Z 1 Z q h (t, ξ, η) Tij `3ξ 2 1, 5 2 `3η 2 1 o φ (m) (ξ, η) φ (n) (ξ, η) dξ dη = δ mn 6X k=1 Q (k) ij (t) φ(k) (ξ, η) J.A. Rossmanith RMMC /35
17 DG-FEM on Cartesian grids Semi-discrete form: d dt Q(l) ij = N (l) ij F (l) ij x G(l) ij y Internal and flux difference parts in the semi-discrete form: N (l) ij = 1 Z 1 Z 1» x φ(l),ξ f(qh ) + 1 y φ(l),η g(q h ) dξ dη,» Z F (l) 1 1 ξ=1 ij = φ (l) f(q h ) dη 2 1 ξ= 1» Z G (l) 1 1 η=1 ij = φ (l) g(q h ) dξ 2 1 η= 1 Integrals approximated via Gaussian quadrature rules Surface integrals: 1D GQ with N points Volume integrals: 2D tensor product GQ with (N 1) (N 1) points J.A. Rossmanith RMMC /35
18 Moment limiters [Krivodonova, 2007] In multi-d, moment limiters become more complicated For example, with a third-order scheme: Limit highest-order first: Q (5) ij Q (4) ij = minmod Q (6) ij Q (4) ξ 2 Q (5) ξη Q (6) η 2 Q (2) ξ Q (1) Q (3) η Q (4) ij, + x Q (2) ij, x Q (2) ij = minmod Q (5) ij, + y Q (2) ij, y Q (2) ij, + x Q (3) ij, x Q (3) ij = minmod If anything happened, go to next order: = minmod Q (2) ij Q (3) ij = minmod Q (6) ij, + y Q (3) ij, y Q (3) ij Q (2) ij, + x Q (1) ij, x Q (1) ij Q (3) ij, + y Q (1) ij, y Q (1) ij J.A. Rossmanith RMMC /35
19 Artificial viscosity [Persson & Peraire, 2006] q,t + F(q) = (ε B(q) v) v = q Cell-by-cell smoothness indicator: q h = K(N) X l=1 Q (l) φ (l), ˆq h = K(N 1) X l=1 Q (l) φ (l) S := qh ˆq h, q h ˆq h, S h 2N (smooth solutions) q h, q h ( 0 if S < TOL ε := ε 0 if S TOL Local DG method for elliptic & parabolic PDEs: Multiply each equation by same test function, φ (l), and 1 int-by-parts Difficulties: time-step restriction due to parabolic part, parameters J.A. Rossmanith RMMC /35
20 Outline 1 1D DG-FEM 2 2D Cartesian DG-FEM 3 2D mesh generation 4 2D unstructured DG-FEM J.A. Rossmanith RMMC /35
21 Triangular elements p(i,1:2) := x and y coordinates of i th node t(j,1:3) := list of 3 node indeces that make up element j All other info about nodes, elements, & edges can be derived from this Example: t(5,1) = 7, t(5,2) = 25, t(5,3) = 18 J.A. Rossmanith RMMC /35
22 Mesh generation Given points, how do we form a triangulation? Mesh quality Minimum angle: θ p «T θ T := min, τ p=1,2,3 60 h := min θ T, 0 τ h 1 T Delaunay: for fixed point locations, this triangulation maximizes τ h Delaunay triangulation Each point gets region of influence based on distance: Voronoi diagram H pq := {x : p x q x } = region p = H pq q This diagram is unique, area of overlap is zero, partitions R 2 Delaunay triangulation is dual to Voronoi: only connect if share edge Voronoi diagram is unique, Delaunay tri. is unique up to degeneracies Degeneracy: 4+ points are co-circular = resolve with coin flip J.A. Rossmanith RMMC /35
23 Mesh generation Given points, how do we form a triangulation? Delaunay triangulation J.A. Rossmanith RMMC /35
24 Mesh generation Given points, how do we form a triangulation? Delaunay triangulation J.A. Rossmanith RMMC /35
25 Mesh generation Given points, how do we form a triangulation? Delaunay triangulation J.A. Rossmanith RMMC /35
26 Mesh generation Given points, how do we form a triangulation? Delaunay triangulation J.A. Rossmanith RMMC /35
27 1D DG-FEM 2D Cartesian DG-FEM 2D mesh generation 2D unstructured DG-FEM Mesh generation Given Ω, how do we form a triangulation? Mesh generator of [Persson & Strang, 2004] Define boundary via 8 > < Φ(x) = + > : 0 signed distance function: if if if x Ω x /Ω, x Ω k Φ(x)k2 = 1 x R2 Start with equi-distributed points in Ω and triangulate J.A. Rossmanith RMMC /35
28 1D DG-FEM 2D Cartesian DG-FEM 2D mesh generation 2D unstructured DG-FEM Mesh generation Given Ω, how do we form a triangulation? Mesh generator of [Persson & Strang, 2004] Spring dynamics: spring const. desired h, only allow repulsion Points that leave Ω are projected onto Ω via x x Φ(x) Φ(x) Rinse and repeat, occasionally retriangulating J.A. Rossmanith RMMC /35
29 Outline 1 1D DG-FEM 2 2D Cartesian DG-FEM 3 2D mesh generation 4 2D unstructured DG-FEM J.A. Rossmanith RMMC /35
30 DG-FEM on unstructured grids Map each element to canonical triangle 1 3, 1 «2 3 3, 1 «3 1 3, 2 «3 x = 1 (x1 + x2 + x3) + ξ (x2 x1) + η (x3 x1) 3 y = 1 (y1 + y2 + y3) + ξ (y2 y1) + η (y3 y1) 3 Basis functions (before Gram-Schmidt orthonormalization): φ (l) = 1, ξ, η, ξ 2, ξη, η 2, ξ 3, ξ 2 η, ξη 2, η 3,... J.A. Rossmanith RMMC /35
31 DG-FEM on unstructured grids Galerkin expansion: q h (t, x) = k(k+1)/2 X n=1 Q (l) (t) φ (l) (x) T start with q,t + F(q) = 0 and obtain semi-discrete weak-form: d dt Q(l) = 1 Z φ (l) F(q) dx 1 I φ (l) F(q) ds T T T T {z } {z } Interior Edge Interior: Gauss quad, Edge: approx Riemann soln, then quadrature J.A. Rossmanith RMMC /35
32 Gaussian quadrature [D.A. Dunavant, 1985] J.A. Rossmanith RMMC /35
33 Artificial viscosity [Persson & Peraire, 2006] q,t + F(q) = (ε B(q) v) v = q Cell-by-cell smoothness indicator: q h = K(N) X l=1 Q (l) φ (l), ˆq h = K(N 1) X l=1 Q (l) φ (l) S := qh ˆq h, q h ˆq h, S h 2N (smooth solutions) q h, q h ( 0 if S < TOL ε := ε 0 if S TOL Local DG method for elliptic & parabolic PDEs: Multiply each equation by same test function, φ (l), and 1 int-by-parts Difficulties: time-step restriction due to parabolic part, parameters J.A. Rossmanith RMMC /35
34 Space/time methods Basic idea Take space-time Galerkin ansatz: q h (t, x) T = NX Q (k) φ (k) (t, x) k=1 Weak formulation: Q (l), n+1 = ±Q (l), n + 1 Z t n+1 Z h i φ (l),t q + φ (l) F(q) dx dt T t n T 1 Z t n+1 I φ (l) F(q) ds dt T t n T J.A. Rossmanith RMMC /35
35 Space/time methods Entropy variables Space/time DG-FEM in entropy variables: Find v h V h such that with B DG(v, w) = B DG(v h, w h ) = 0, w h V h N 1 X n=0 Z + X K T + X Z K T X Z K T I n K Z K Entropy stability: Z Z U(q(v h (t N, x))) dx Ω I n Z K (q(v) w,t + f i(v) w,xi ) dx dt w(x ) h(v(x ), v(x +); n) ds dt n+1 `w(t ) q(v(t n+1 )) w(t n +) q(v(t n )) dx Ω U(q(v h (t 0, x)) dx! J.A. Rossmanith RMMC /35
36 Lax-Wendroff time-stepping Basic idea Another alternative to RK is to do a Taylor series in time This turns out to be useful for local time-stepping Taylor series expansion: q,t + F(q) = ψ(q) q(t + t, x, y) = q + t q,t + t2 2 q,tt + O( t3 ) = q tˆ F Ψ F := F t 2 F,q { F ψ} + O( t2 ) Ψ := ψ t 2 ψ,q { F ψ} + O( t2 ). Weak formulation: n o n+1 n o n ZZ Q (l) i = Q (l) t i + φ (l) F n dx T i T i t I φ (l) F n n ds + t ZZ φ (l) Ψ n dx T i T i T i T i J.A. Rossmanith RMMC /35
37 Lax-Wendroff time-stepping Asynchronous steps Each element has its own time-step: 1 Sort each element by their current time t i 2 Update the element that is lagging furthest behind 3 Apply flux corrections to maintain global conservation 4 Place the updated element in the correct location of the ordered list 5 Rinse and repeat J.A. Rossmanith RMMC /35
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